Locale pullback via dcpos

1
Locale pullback via dcpos

• Dr Christopher Townsend
• (Open University)

2
Main Idea

• THESIS
• When changing base it is only really the
  directed joins that need to be modelled/worried
  about. All the rest of the finitary data takes
  care of itself.

• Study locales only, i.e. frames. I.e. the data is
  finite meets and arbitrary joins. Equivalently
  finite meets, finite joins and DIRECTED joins.
• The definition of geometric morphism (suggests
  atleast) that the finitary structure is preserved.

3
Technical Aims

• Given fE -gt E, a geometric morphism. Then the
  direct image functor preserves dcpo structure.
  I.e. fdcpoE -gt dcpoE is well defined.
• Further fdcpoE -gt dcpoE has a left adjoint f.
  
• In any topos Dlat(dcpo)Frames. Where we look at
  order-internal distributive lattices in the
  order enriched category dcpo.
• The left adjoint f restricts to a functor
  Dlat(dcpoE)-gt Dlat(dcpoE) left adjoint to fFrE
  -gt FrE That is locale pullback.

4
What is known already?

• This trick has been done by Joyal and Tierney
  already with suplattices -

f
Joyal and Tierney 84
dcpoE
dcpoE
NEW!
f
f
supE
supE
Frames as ring objects in sup
f
Frames as order-internal dlats in dcpo
f
FrE
FrE
f
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Is fdcpoE -gt dcpoE well defined?

• YES. Use external definition of dcpo.

x fiber directed iff x-1(j) directed for all j.
(I,J posets.)
As with sup lattices -
For every fiber directed xI-gtJ, the map xPosE
(J,A)-gt PosE(I,A) has a left adjoint (and
Beck-Chevalley holds).
There exists VIdlA-gtA left adjoint to
A-gtIdlA
Internal Definition of dcpo
External Definition

• Then unravel the adjunction of the geometric
  morphism with the external definition to prove f
  is well defined. This works as fiber directedness
  is stable under the inverse image. (Known?)

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Defining fdcpoE -gt dcpoE
TRICK Use presentations. For every dcpo A, there
exists posets G and R and dcpo maps e1 and e2
such that
e1
is a coequalizer
A
Idl(R)
Idl(G)
e2
Dcpo coequalizer well defined? Folklore, or adapt
Johnstone Vickers 91
f
f e1
f A
Idl(f R)
Idl(f G)
defines f A.
f e2
Note e1 and e2 are suitably geometric and so f
e1 and f e1 well defined
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Frames as Order-Internal DLats
DEFINITION For any order enriched category C
(with lax products), an object X is an
order-internal meet semilattice iff !X-gt1 and
?Xgt-gtXxX have right adjoints.
in other words, iff finite complete wrt to the
order enrichment

• Define order-internal distributive lattice in the
  standard way from this. Then -

THEOREM FrDlat(dcpo)
Proof A in Dlat(dcpo), then